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In order to choose the best number of underlying factors for my data using factor analysis, I decided to use the tutorial outlined in scikit-learn's documentation.

Running cross_val_score(fa, X) outputs a score (usually a negative number). What is this score actually measuring? Any references with your answer would be much appreciated!

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The score you get is determined by the estimator you use. In your case this is Factor Analysis. Looking at the documentation of the algorithm, you can find the explanation:

FactorAnalysis performs a maximum likelihood estimate of the so-called loading matrix, the transformation of the latent variables to the observed ones, using expectation-maximization (EM).

and the reference:

score(X[, y])   Compute the average log-likelihood of the samples

So your number is the logarithm of the maximum likelihood estimate of the factor loading matrix.

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